I am encountering non-local (nonlinear) PDEs in my work. To compute stability, I am trying to numerically estimate the spectrum of linearized-but-nonlocal version of the said PDEs.

However, I am not sure if all the lessons from the local PDE spectral theory apply here. So I am looking for a good reference/monograph/review paper which describes the subtle points in nonlocal spectral theory and/or stability analysis, especially issues that do not arise in local PDE theory.

Some examples of stuff I am looking for:

1). One example is Evans function computation, whose theory is only complete for the local case.

2). Does there exist a Strum-Liouville type theory for non-local operators ?